$82$ people attended a baseball game. Everyone there was a fan of either the home team or the away team. The number of home team fans was $93$ less than $4$ times the number of away team fans. How many home team and away team fans attended the game?
Solution: Let $x$ equal the number of home team fans and $y$ equal the number of away team fans. The system of equations is then: ${x+y = 82}$ ${x = 4y-93}$ Solve for $x$ and $y$ using substitution. Since $x$ has already been solved for, substitute ${4y-93}$ for $x$ in the first equation. ${(4y-93)}{+ y = 82}$ Simplify and solve for $y$ $ 4y-93 + y = 82 $ $ 5y-93 = 82 $ $ 5y = 175 $ $ y = \dfrac{175}{5} $ ${y = 35}$ Now that you know ${y = 35}$ , plug it back into ${x = 4y-93}$ to find $x$ ${x = 4}{(35)}{ - 93}$ $x = 140 - 93$ ${x = 47}$ You can also plug ${y = 35}$ into ${x+y = 82}$ and get the same answer for $x$ ${x + }{(35)}{= 82}$ ${x = 47}$ There were $47$ home team fans and $35$ away team fans.